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Theorem isoml 37703
Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isoml.b 𝐡 = (Baseβ€˜πΎ)
isoml.l ≀ = (leβ€˜πΎ)
isoml.j ∨ = (joinβ€˜πΎ)
isoml.m ∧ = (meetβ€˜πΎ)
isoml.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
isoml (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐾,𝑦
Allowed substitution hints:   ∨ (π‘₯,𝑦)   ≀ (π‘₯,𝑦)   ∧ (π‘₯,𝑦)   βŠ₯ (π‘₯,𝑦)

Proof of Theorem isoml
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6843 . . . 4 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
2 isoml.b . . . 4 𝐡 = (Baseβ€˜πΎ)
31, 2eqtr4di 2795 . . 3 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
4 fveq2 6843 . . . . . . 7 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
5 isoml.l . . . . . . 7 ≀ = (leβ€˜πΎ)
64, 5eqtr4di 2795 . . . . . 6 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
76breqd 5117 . . . . 5 (π‘˜ = 𝐾 β†’ (π‘₯(leβ€˜π‘˜)𝑦 ↔ π‘₯ ≀ 𝑦))
8 fveq2 6843 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = (joinβ€˜πΎ))
9 isoml.j . . . . . . . 8 ∨ = (joinβ€˜πΎ)
108, 9eqtr4di 2795 . . . . . . 7 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = ∨ )
11 eqidd 2738 . . . . . . 7 (π‘˜ = 𝐾 β†’ π‘₯ = π‘₯)
12 fveq2 6843 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (meetβ€˜π‘˜) = (meetβ€˜πΎ))
13 isoml.m . . . . . . . . 9 ∧ = (meetβ€˜πΎ)
1412, 13eqtr4di 2795 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (meetβ€˜π‘˜) = ∧ )
15 eqidd 2738 . . . . . . . 8 (π‘˜ = 𝐾 β†’ 𝑦 = 𝑦)
16 fveq2 6843 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (ocβ€˜π‘˜) = (ocβ€˜πΎ))
17 isoml.o . . . . . . . . . 10 βŠ₯ = (ocβ€˜πΎ)
1816, 17eqtr4di 2795 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (ocβ€˜π‘˜) = βŠ₯ )
1918fveq1d 6845 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((ocβ€˜π‘˜)β€˜π‘₯) = ( βŠ₯ β€˜π‘₯))
2014, 15, 19oveq123d 7379 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑦(meetβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘₯)) = (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))
2110, 11, 20oveq123d 7379 . . . . . 6 (π‘˜ = 𝐾 β†’ (π‘₯(joinβ€˜π‘˜)(𝑦(meetβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘₯))) = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))
2221eqeq2d 2748 . . . . 5 (π‘˜ = 𝐾 β†’ (𝑦 = (π‘₯(joinβ€˜π‘˜)(𝑦(meetβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘₯))) ↔ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯)))))
237, 22imbi12d 345 . . . 4 (π‘˜ = 𝐾 β†’ ((π‘₯(leβ€˜π‘˜)𝑦 β†’ 𝑦 = (π‘₯(joinβ€˜π‘˜)(𝑦(meetβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘₯)))) ↔ (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
243, 23raleqbidv 3320 . . 3 (π‘˜ = 𝐾 β†’ (βˆ€π‘¦ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑦 β†’ 𝑦 = (π‘₯(joinβ€˜π‘˜)(𝑦(meetβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘₯)))) ↔ βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
253, 24raleqbidv 3320 . 2 (π‘˜ = 𝐾 β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘¦ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑦 β†’ 𝑦 = (π‘₯(joinβ€˜π‘˜)(𝑦(meetβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘₯)))) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
26 df-oml 37644 . 2 OML = {π‘˜ ∈ OL ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘¦ ∈ (Baseβ€˜π‘˜)(π‘₯(leβ€˜π‘˜)𝑦 β†’ 𝑦 = (π‘₯(joinβ€˜π‘˜)(𝑦(meetβ€˜π‘˜)((ocβ€˜π‘˜)β€˜π‘₯))))}
2725, 26elrab2 3649 1 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 β†’ 𝑦 = (π‘₯ ∨ (𝑦 ∧ ( βŠ₯ β€˜π‘₯))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  occoc 17142  joincjn 18201  meetcmee 18202  OLcol 37639  OMLcoml 37640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-oml 37644
This theorem is referenced by:  isomliN  37704  omlol  37705  omllaw  37708
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